ANNALS OF
MATHEMATICS
The perverse filtration and the Lefschetz
hyperplane theorem
By Mark Andrea A. de Cataldo and Luca Migliorini
SECOND SERIES, VOL. 171, NO. 3
May, 2010
anmaah
Annals of Mathematics, 171 (2010), 2089–2113
The perverse filtration and the Lefschetz
hyperplane theorem
By M ARK A NDREA A. DE C ATALDO and L UCA M IGLIORINI
Abstract
We describe the perverse filtration in cohomology using the Lefschetz hyperplane theorem.
1.
Introduction
2.
Notation
3.
The perverse and flag spectral sequences
3.1. .K; P /
3.2. Flags
3.3. .K; F; G; ı/
3.4. The graded complexes associated with .K; P; F; G; ı/
3.5. .R€.Y; K/; P; F; ı/ and .R€c .Y; K/; P; G; ı/
3.6. The perverse and flag spectral sequences
3.7. The shifted filtration and spectral sequence
4.
Results
4.1. The results over an affine base
4.2. The results over a quasi projective base
5.
Preparatory material
5.1. Vanishing results
5.2. Transversality, base change and choosing good flags
5.3. Two short exact sequences
5.4. The forget-the-filtration map
5.5. The canonical lift of a t-structure
5.6. The key lemma on bifiltered complexes
6.
Proof of the results
6.1. Verifying the vanishing (32) for general flags
The first named author was partially supported by N.S.F. The second named author was partially
supported by GNSAGA and PRIN 2007 project “Spazi di moduli e teoria di Lie”.
2089
2090
6.2.
6.3.
7.
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
Proofs of Theorems 4.1.1, 4.1.2, 4.1.3 and 4.2.1
Resolutions in D b .PY /
Applications
References
1. Introduction
In this paper we give a geometric description of the middle perverse filtration
on the cohomology and on the cohomology with compact supports of a complex
with constructible cohomology sheaves of abelian groups on a quasi projective variety. The description is in terms of restriction to generic hyperplane sections and it is
somewhat unexpected, especially if one views the constructions leading to perverse
sheaves as transcendental and hyperplane sections as more algebro-geometric.
The results of this paper are listed in Section 4, and hold for a quasi projective
variety. For the sake of simplicity, we describe here the case of the cohomology of
an n-dimensional affine variety Y AN with coefficients in a complex K.
The theory of t -structures endows the (hyper)cohomology groups H .Y; K/
with a canonical filtration P , called the perverse filtration,
P p H.Y; K/ D Im fH.Y; p
p K/
! H.Y; K/g;
which is the abutment of the perverse spectral sequence. Let
Y D fY Y
1
Y
ng
be a sequence of closed subvarieties; we call this data an n-flag. Basic sheaf theory endows H.Y; K/ with the so-called flag filtration F , abutment of the spectral
p;q
sequence E1 D H pCq .Yp ; Yp 1 ; KjYp / H) H .Y; K/. We have F p H.Y; K/
D Ker fH.Y; K/ ! H.Yp 1 ; KjYp 1 /g. For an arbitrary n-flag, the perverse and
flag filtrations are unrelated.
In terms of filtrations, the main result of this paper is that if the n-flag is
obtained using n hyperplane sections in sufficiently general position, then
(1)
P p H j .Y; K/ D F pCj H j .Y; K/:
More precisely, we construct a complex R€.Y; K/ endowed with two filtrations
P and F and we prove (Theorem 4.1.1) that there is a natural isomorphism in the
filtered derived category DF .Ab/ of abelian groups
(2)
.R€.Y; K/; P / D .R€.Y; K/; Dec.F //;
where Dec.F / is the shifted filtration associated with F . Then (1) follows from (2).
Our methods seem to break down in the non quasi projective case and also
for other perversities.
THE LEFSCHETZ HYPERPLANE THEOREM
2091
The constructions and results are amenable to mixed Hodge theory. We offer
the following application: let f W X ! Y be any map of algebraic varieties, Y
be quasi projective and C be a bounded complex with constructible cohomology
sheaves on X: Then the perverse Leray spectral sequences can be identified with
suitable flag spectral sequences on X: In the special case when K D QX ; we obtain
the following result due to M. Saito: the perverse spectral sequences for H.X; Q/
and Hc .X; Q/; are spectral sequences of mixed Hodge structures. Further Hodgetheoretic applications concerning the decomposition theorem are mentioned in Remark 7.0.5 and will appear in [6].
The isomorphism (2) lifts to the bounded derived category D b .PY / of perverse sheaves with rational coefficients. This was the basis of the proof of our
results in an earlier version of this paper. The present formulation, which shortcircuits D b .PY /, is based on the statement of Proposition 5.6.1 which has been
suggested to us by an anonymous referee. We are deeply grateful for this suggestion. The main point is that a suitable strengthening of the Lefschetz hyperplane
theorem yields cohomological vanishings for the bifiltered complex .R€.Y; K/;
P; F / which yield (2). These vanishings are completely analogous to the ones
occurring for topological cell complexes and, for example, one can fit the classical
Leray spectral sequence of a fiber bundle in the framework of this paper.
The initial inspiration for this work comes from Arapura’s paper [1], which
deals with the standard filtration, versus the perverse one. In this case, the flag
has to be special: it is obtained by using high degree hypersurfaces containing the
bad loci of the ordinary cohomology sheaves. The methods of this paper are easily
adapted to that setting; see [7].
The fact that the perverse filtration is related to general hyperplane sections
confirms, in our opinion, the more fundamental role played by perverse sheaves
with respect to ordinary sheaves. The paper [1] has also directed us to the beautiful
[17] and the seminal [4]. The influence on this paper of the ideas contained in [4],
[17] is hard to overestimate.
2. Notation
A variety is a separated scheme of finite type over the field of complex numbers C. A map of varieties is a map of C-schemes. The results of this paper hold
for sheaves of R-modules, where R is a commutative ring with identity with finite
global dimension, e.g. R D Z; R a field, etc. For the sake of exposition we work
with R D Z, i.e. with sheaves of abelian groups.
The results of this paper hold, with routine adaptations of the proofs, in the
case of varieties over an algebraically closed field and étale sheaves with the usual
coefficients: Z= l m Z; Zl ; Ql , Zl ŒE, Ql ŒE (E Ql a finite extension) and Ql :
We do not discuss further these variants, except to mention that the issue of
stratifications is addressed in [5, 2.2 and 6]. The term stratification refers to an
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MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
algebraic Whitney stratification [14]. Recall that any two stratifications admit a
common refinement and that maps of varieties can be stratified.
Given a variety Y , there is the category DY D DY .Z/ which is the full
subcategory of the derived category of the category ShY of sheaves of abelian
groups whose objects are the bounded complexes with constructible cohomology
sheaves, i.e. bounded complexes K whose cohomology sheaves Hi .K/, restricted
to the strata of a suitable stratification † of Y , become locally constant with fiber
a finitely generated abelian group. For a given †; a complex with this property is
called †-constructible.
Given a stratification † of Y , there are the full subcategories DY† DY of
complexes which are †-constructible. Given a map f W X ! Y of varieties, there
are the usual four functors .f ; Rf ; RfŠ ; f Š /. By abuse of notation, we denote
Rf and RfŠ simply by f and fŠ . The four functors preserve stratifications; i.e. if
0
0
f W .X; †0 / ! .Y; †/ is stratified, then f ; fŠ W DX† ! DY† and f ; f Š W DY† ! DX† :
The abelian categories ShY and Ab WD Shpt have enough injectives. The right
derived functor of global sections is denoted R€.Y; /: Hypercohomology groups
are denoted simply by H.Y; K/: Similarly, we have R€c .Y; / and Hc .Y; K/:
We consider only the middle perversity t-structure on DY [5]. The truncation
j
functors are denoted pi W DY ! pDYi , pj W DY ! pDY , the heart PY WD
pD 0 \ pD 0 is the abelian category of perverse sheaves on Y and we denote the
Y
Y
perverse cohomology functors pHi WD p0 ı p0 ı Œi  W DY ! PY .
The perverse t -structure is compatible with a fixed stratification, i.e. truncations preserve †-constructibility and we have pHi W DY† ! P†
Y , etc.
In this paper, the results we prove in cohomology have a counterpart in cohomology with compact supports. If we employ field coefficients, then middle
perversity is preserved by duality and the results in cohomology are equivalent
to the ones in cohomology with compact supports by virtue of Poincaré-Verdier
Duality.
Due to the integrality of the coefficients, middle perversity is not preserved
by duality; see [5, 3.3]. However, we can prove the results in cohomology and
in compactly supported cohomology using the same techniques. For expository
reasons, we often emphasize cohomology.
Filtrations, on groups and complexes, are always finite, i.e. F i K D K for
i 0 and F j K D 0 for j 0, and decreasing, i.e. F i K F i C1 K: We say that
p
F has type Œa; b, for a b 2 Z; if GrF K ' 0 for every i … Œa; b:
A standard reference for the filtered derived category DF .A/ of an abelian
category is [15]. Useful complements can be found in [5, 3] and in [4, Appendix].
We denote the filtered version of DY by DY F . The objects are filtered complexes
.K; F /, with K 2 DY : This is a full subcategory of D b F .ShY /:
We denote a “canonical” isomorphism with the symbol “D :”
2093
THE LEFSCHETZ HYPERPLANE THEOREM
3. The perverse and flag spectral sequences
In this paper, we relate the perverse spectral sequences with certain classical
objects that we call flag spectral sequences.
In order to do so, we exhibit these spectral sequences as the ones associated
with a collection of filtered complexes of abelian groups. These, in turn, arise by
taking the global sections of (a suitable injective model of) the complex K endowed
with the filtrations P; F; G and ı which we are about to define.
In this section, starting with a variety Y and a complex K 2 DY , we construct
the multi-filtered complex .K; P; F; G; ı/ and we list its relevant properties. By
passing to global sections, we identify the ensuing spectral sequences of filtered
complexes with the perverse and flag ones.
3.1. .K; P /. The system of truncation maps ! p p K ! p pC1 K
! : : : is isomorphic in DY to a system of inclusion maps ! P p K 0 ! P p 1 K 0
p
! : : : , where the filtered complex .K 0 ; P / is of injective type, i.e. all GrP K 0 , hence
all P p K 0 and K 0 , have injective entries; see [5, 3.1.2.7]. The filtered complex
.K 0 ; P / is well-defined up to unique isomorphism in the filtered DY F by virtue of
[5, Prop. 3.1.4.(i)] coupled with the second axiom, “Hom 1 D 0,” of t -structures.
We replace K with K 0 and obtain .K; P /: In particular, from now on, K is injective.
3.2. Flags. The smooth irreducible projective variety F .N; n/ of n-flags on
the N -dimensional projective space PN parametrizes linear n-flags F D fƒ 1 ƒ n g, where ƒ p PN is a codimension p linear subspace.
A linear n-flag F on PN is said to be general if it belongs to a suitable Zariski
dense open subset of the variety of flags F .N; n/: We say that a pair of flags is
general if the same is true for the pair with respect to F .N; n/ F .N; n/: In this
paper, this open set depends on the complex K and on the fixed chosen embedding
Y PN . We discuss this dependence in Section 5.2.
A linear n-flag F on PN gives rise to an n-flag on Y PN , i.e. an increasing
sequence of closed subvarieties of Y :
(3)
Y D Y .F/ W
We set Y n
beddings:
(4)
1
Y D Y0 Y
1
Y
n;
Yp WD ƒp \ Y:
WD ∅ and we have the (resp., closed, open and locally closed) em-
ip W Yp ! Y;
jp W Y n Yp
1
! Y;
kp W Yp n Yp
1
! Y:
Let h W Z ! Y be a locally closed embedding. There are the exact functor
. /Z D hŠ h . /, which preserves c-softness, and the left exact functor €Z , which
preserves injectivity and satisfies HZ .Y; K/ D H.Y; R€Z K/; see [16]. If h is
closed, then R€Z D hŠ hŠ D h hŠ I and, since K is injective, R€Z K D €Z K: If
2094
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
Z 0 Z is closed, then we have the distinguished triangle R€Z 0 K ! R€Z K !
C
R€Z Z 0 K ! which, again by the injectivity of K, is the triangle associated with
the exact sequence 0 ! €Z 0 K ! €Z K ! €Z Z 0 K ! 0:
3.3. .K; F; G; ı/. We have constructed .K; P / of injective type. Let Y PN
be an embedding of the quasi projective variety Y . Let F; F0 be two, possibly
identical, linear n-flags on PN with associated flags, Y and Z on Y: We denote
the corresponding maps (4) by i 0 ; j 0 ; k 0 :
We define the three filtrations F; G and ı on K. They are well-defined, up to
unique isomorphism, in the filtered DY F:
The flag filtration F D FY D FY .F/ , of type Œ n; 0, is defined by setting
WD KY Yp 1 :
F pK
0 KY
(5)
Y
1
: : : KY
Yp
1
KY
Y
n
K:
The flag filtration G D GZ D GZ .F0 / , of type Œ0; n, is defined by setting
WD €Z p K W
GpK
(6)
0 €Z
n
K : : : €Z
p
K €Z
1
K K:
The flag filtration ı D ı.FY .F/ GZ .F0 / /, of type Œ n; n, is the diagonal filP
tration defined by ı p K D i Cj Dp F i K \ G j K:
Note that one does not need injectivity to define the filtrations. However,
without this assumption, the resulting filtration G and ı would not be canonically
defined in DY F: Moreover, injectivity yields €Z K D R€Z K for every locally
closed Z Y , a fact we use throughout without further mention.
p
3.4. The graded complexes associated with .K; P; F; G; ı/. Recall that Grı
j
j
j
D ˚i Cj Dp GriF GrG and that the Zassenhaus Lemma implies GriF GrG D GrG GriF :
Since the formation of F is an exact functor, the formation of G is exact when
applied to injective sheaves, and injective sheaves are c-soft, we have
(7)
p
GrG K;
j
p
GrG GrP
p
GriF GrP K;
p
Grı K;
GrF K;
(8)
j
p
GrP K;
GriF GrG GrP K;
p
p
are injective,
j
GriF GrG K;
p
Graı GrP K
are c-soft.
In particular, P p K; G p K and P p K \ G j K are injective. We have the analogous c-softness statement for (8), e.g. the F i K \ G j \ P p K are c-soft. By
construction (3.1), the filtration P splits in each degree, and the formation of F
and G is compatible with direct sums. Hence, we have the following list of natural
isomorphisms:
2095
THE LEFSCHETZ HYPERPLANE THEOREM
p
1) GrP K D pH
p .K/Œp,
p
2) GrF K D KYp
Yp
p
3) GrG K D €Z
p
Z
D kp Š kp K,
1
p 1
K D k0
p k
j
j
4) GrG GriF K DGriF GrG K D .€Z
p
5) Grı K D ˚i Cj Da .€Z
p
6) GriF GrP D . pH
j
j
Z
j
j
1
. pH
8) GriF GrG GrP K D .€Z
j
Z
j
9)
j
Z
p
p
Graı GrP K
K/Yi
p .K/Œp/
Yp Yp
p
7) GrG GrP D €Z
D ˚i Cj Da .€Z
j
Z
j
1
0Š
1
p K.
j
1
Z
1
Yi
D.k 0
1
j k
0Š
j K/Yi Yi
1
.
.
.
. pH
j
K/Yi
Yi
p .K/Œp/
j
1
1
D k0
j k
0Š
j
p .K/Œp//
Yi Yi
. pH
pH p .K/Œp.
1
.
p .K/Œp//
Yi Yi
1
.
Remark 3.4.1. If the pair of flags is general, then (cf. §5.2, or [4, Complement
to §3])
.€Z
j
Z
j
1
K/Yi
Yi
1
D €Z
j
Z
j
1
.KYi
Yi
1
In general, the two sides differ, for the left-hand side is zero on Yi
/:
1:
3.5. .R€.Y; K/; P; F; ı/ and .R€c .Y; K/; P; G; ı/. Since K is injective, we
have R€.Y; K/ D €.Y; K/, R€c .Y; K/ D €c .Y; K/: We keep “R” in the notation.
By applying the left exact functors € and €c , we obtain the multi-filtered
complexes of abelian groups
.R€.Y; K/; P; F; ı/;
(9)
.R€c .Y; K/; P; G; ı/;
by setting, for example, P p R€.Y; K/ WD €.Y; P p K/, etc.
Since injective sheaves and c-soft sheaves are € and €c -injective, we have
p
p
(10)
R€.Y; pH
p
.K/Œp/ D €.Y; GrP K/ D GrP €.Y; K/;
(11)
R€c .Y; pH
p
.K/Œp/ D €c .Y; GrP K/ D GrP €c .Y; K/;
p
p
with analogous formulæ for the following graded objects
(12)
p
p
j
p
j
p
b
b
b
b
GrF ; GrG ; GraF GrP
; GraG GrP
; GriF GrG ; Grı ; GriF GrG GrP
; Grı GrP
:
Remark 3.5.1. Though the formation of F does not preserve injectivity, one
can always take filtered injective resolutions. In that case, the resulting F p K is not
exactly KY Yp 1 , etc., but rather an injective resolution of it. This would allow us
to drop the mention of c-softness. On the other hand, the F -construction is exact
and formulæ like the ones in Section 3.4 are readily proved.
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MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
3.6. The perverse and flag spectral sequences. With the aid of Sections 3.4,
3.5 it is immediate to recognize the E1 -terms of the spectral sequences associated
with the filtered complexes .R€.Y; K/; P; F; ı/ and .R€c .Y; K/; P; G; ı/:
Definition 3.6.1 (Perverse spectral sequence and filtration). The perverse spectral sequence for H.Y; K/ is the spectral sequences of the filtered complexes
.R€.Y; K/; P /:
(13)
p;q
E1
D H 2pCq .Y; pH
p
.K// H) H .Y; K/
and the abutment is the perverse filtration P on H .Y; K/ defined by
(14)
P p H .Y; K/ D Im fH .Y; p
p K/
! H .Y; K/g;
similarly, for Hc .Y; K/ using .R€c .Y; K/; P /:
Let f W X ! Y be a map of algebraic varieties and C 2 DX :
Definition 3.6.2. The perverse Leray spectral sequences for H.X; C / (resp.
Hc .X; C /) are the corresponding perverse spectral sequences on Y for K WD f C
(resp. K WD fŠ C ).
Let Y PN be an embedding of the quasi projective variety Y , F; F0 be two
linear flags on PN and Y and Z be the corresponding flags on Y .
Definition 3.6.3 (Flag spectral sequence and filtration (F -version)). The F
flag spectral sequence associated with Y is the spectral sequence associated with
the filtered complex .R€.Y; K/; F /:
(15)
p;q
E1
D H pCq .Y; KYp
Yp
1
/ H) H .Y; K/
and its abutment is the flag filtration F D FY on H .Y; K/ defined by
(16)
F p H .Y; K/ D KerfH .Y; K/ ! H .Yp
1 ; KjYp
1
/g:
Definition 3.6.4 (Flag spectral sequence and filtration (G-version)). The G
flag spectral sequence associated with Z is the spectral sequence associated with
the filtered complex .R€c .Y; K/; G/:
(17)
p;q
E1
D HcpCq .Y; k
Š
p k p K/
H) Hc .Y; K/
and its abutment is the flag filtration G D GZ on Hc .Y; K/ defined by
(18)
G p Hc .Y; K/ D Imf Hc .Y; €Z
p
K/ ! Hc .Y; K/g:
Definition 3.6.5 (Flag spectral sequence and filtration (ı-version)). The ı flag
spectral sequences associated with .Y ; Z / are the spectral sequences associated
with the filtered complexes .R€.Y; K/; ı/ and .R€c .Y; K/; ı/.
2097
THE LEFSCHETZ HYPERPLANE THEOREM
Remark 3.6.6. We omit displaying these spectral sequences since, due to Remark 3.4.1, they do not have familiar E1 -terms. If the pair of flags is general, or
merely in good position with respect to † and each other (cf. §5.2), then we have
equality in Remark 3.4.1 and the E1 -terms take the following form (we write Hc;Z
for Hc ı R€Z ):
M
p;q
pCq
(19)
E1 D
HZ j Z j 1 .Y; KYi Yi 1 / H) H .Y; K/;
i Cj Dp
(20)
p;q
E1
D
M
pCq
Hc;Z
j
Z
j
1
.Y; KYi
Yi
1
/ H) Hc .Y; K/
i Cj Dp
and their abutments are the flag filtrations ı D ı.Y ; Z / on H.Y; K/ and on
Hc .Y; K/ defined by
M
p (21)
ı H .Y; K/ D Im
HZ j .Y; KY Yi / ! H .Y; K/ ;
i Cj Dp
(22)
M
p ı Hc .Y; K/ D Im
Hc;Z
j
.Y; KY
Yi /
! H .Y; K/ :
i Cj Dp
3.7. The shifted filtration and spectral sequence. We need the notion and basic properties ([10]) of the shifted filtration for a filtered complex .L; F / in an
abelian category. We make the definition explicit in Ab:
The shifted filtration Dec.F / on L is:
n
o
Dec.F /p Ll WD x 2 F pCl K l j dx 2 F pClC1 LlC1 :
The shifted spectral sequence of .L; F / is the one for .L; Dec.F // and we have
(23)
2pCq; p
Dec.F /p H l .L/ D F pCl H l .L/;
Erp;q .L; Dec.F // D ErC1
.L; F /:
4. Results
We prove results for Y quasi projective. The statements and the proofs are
more transparent when Y is affine. We state and prove the results in the affine
case first. The multi-filtered complexes of abelian groups .R€.Y; K/; P; F; ı/ and
.R€c .Y; K/; P; G; ı/, which give rise to the spectral sequences and filtrations we
are interested in, are defined in Section 3.
4.1. The results over an affine base. In this section Y is affine of dimension
n and K 2 DY . Let Y PN be a fixed embedding and F; F0 be a pair of linear
n-flags on PN .
T HEOREM 4.1.1 (Perverse filtration on cohomology for affine varieties). Let
F be general. There is a natural isomorphism in the filtered derived category
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MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
DF .Ab/:
.R€.Y; K/; P / ' .R€.Y; K/; Dec.F //
identifying the perverse spectral sequence with the shifted flag spectral sequence
so that
P p H l .Y; K/ D F pCl H l .Y; K/ D KerfH l .Y; K/ ! H l .YpCl
1 ; KjYpCl
1
/g:
T HEOREM 4.1.2 (Perverse filtration on Hc and affine varieties). Let F0 be
general. There is a natural isomorphism in the filtered derived category DF .Ab/:
.R€c .Y; K/; P / ' .R€c .Y; K/; Dec.G//
identifying the perverse spectral sequence with the shifted flag spectral sequence
so that
P p Hcl .Y; K/ D G pCl Hcl .Y; K/ D ImfHcl .Y; R€Z
p l
K/ ! Hcl .Y; K/g:
In what follows, f W X ! Y is an algebraic map, with Y affine, C 2 DX ,
and given a linear n-flag F on PN , we denote by X D f 1 Y the corresponding
pre-image n-flag on X:
T HEOREM 4.1.3 (Perverse Leray and affine varieties). Let F be general. The
perverse Leray spectral sequence for H.X; C / is the corresponding shifted X flag
spectral sequence. The analogous statement for Hc .X; C / holds.
Remark 4.1.4. The ı-variants of Theorems 4.1.1, 4.1.2, and 4.1.3 for cohomology and for cohomology with compact supports, hold for the ı filtration as
well, and with the same proof. In this case one requires the pair of flags to be
general.
Remark 4.1.5. Rather surprisingly, the differentials of the perverse (Leray)
spectral sequences can be identified with the differentials of a flag spectral sequence. In turn, these are classical algebraic topology objects stemming from a
filtration by closed subsets, i.e. from the cohomology sequences associated with
the triples .Yp ; Yp 1 ; Yp 2 /:
4.2. The results over a quasi projective base. In this section, Y is a quasi
projective variety of dimension n and K 2 DY :
There are several ways to state and prove generalizations of the results in
Section 4 to the quasi projective case. We thank an anonymous referee for suggesting this line of argument as an alternative to our original two arguments that used
Jouanolou’s trick (as in [1]), and finite and affine Čech coverings. For an approach
via Verdier’s spectral objects see [6].
Let Y be quasi projective, Y PN be a fixed affine embedding and .F; F0 / be
a pair of linear n-flags on PN . The notion of a ı flag spectral sequence is defined
in Definition 3.6.3; see also Remark 3.6.6.
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THE LEFSCHETZ HYPERPLANE THEOREM
T HEOREM 4.2.1 (Quasi projective case via two flags). Let the pair of flags be
general. There are natural isomorphisms in DF .Ab/:
.R€.Y; K/; P / ' .R€.Y; K/; Dec.ı//; .R€c .Y; K/; P / ' .R€c .Y; K/; Dec.ı//
identifying the perverse and the shifted ı flag spectral sequence, inducing the identity on the abutted filtered spaces.
Moreover, if f W X ! Y and C 2 DX are given, then the perverse Leray
spectral sequences coincide with the shifted ı flag spectral sequences associated
with the preimage flags on X:
5. Preparatory material
5.1. Vanishing results.
T HEOREM 5.1.1 (Cohomological dimension of affine varieties). Let Y be
affine and Q 2 PY be a perverse sheaf on Y . Then
H r .Y; Q/ D 0;
for all r > 0;
Hcr .Y; Q/ D 0;
for all r < 0:
Proof. We give several references. The original proof of the first statement is
due to Michael Artin [2], XIV and is valid in the étale context. [14, 2.5] proved
the theorem for intersection homology with compact supports and with twisted
coefficients on a pure-dimensional variety; the reader can translate the results in
intersection cohomology and intersection cohomology with compact supports on
a pure-dimensional variety; a standard devissage argument implies the result for
a perverse sheaf Q on arbitrary varieties; Q is a finite extension of intersection
cohomology complexes with twisted, not necessarily semisimple, coefficients on
the irreducible components. In [5, Th. 4.1.1] the case of H is proved directly; the
case of Hc is proved for field coefficients by invoking duality; however, one can
prove it directly and for arbitrary coefficients. The textbook [16] proves it for Stein
manifolds (see loc. cit., Th. 10.3.8); the general case follows by embedding Y as
a closed subset of an affine space i W Y ! Cn and by applying the statement to the
perverse sheaf i Q.
Let Y be quasi projective. Fix an affine embedding Y PN . Let ƒ; ƒ0 PN
be two hyperplanes, H WD Y \ ƒ Y and j W Y n H ! Y
H W i be the
corresponding open and closed immersions. Note that j Š D j : Similarly, for ƒ0 .
T HEOREM 5.1.2 (Strong Weak Lefschetz). Let Y be quasi projective and Q 2
PY : If ƒ is general, then
H r .Y; jŠ j Š Q/ D 0; 8r < 0;
Hcr .Y; j j Q/ D 0; 8r > 0:
Let .ƒ; ƒ0 / be a general pair. Then jŠ j j0 j 0 Š Q D j0 j 0 Š jŠ j Q and
Š
Š
H r .Y; jŠ j j0 j 0 Q/ D Hcr .Y; j0 j 0 jŠ j Q/ D 0; 8r ¤ 0:
2100
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
Proof. We give several references for the first statement. In [4, Lemma 3.3];
this proof is valid in the étale context. The second statement is observed in [4,
Complement to 3], [14, 2.5]. M. Goresky has informed us that P. Deligne has
also proved this result (unpublished).
We include a sketch of the proof of this result, following [4], in Section 5.2,
where we also complement the arguments in [4] needed in the sequel of the paper.
Remark 5.1.3. Since j Š D j , we may reformulate the first statement of Theorem 5.1.2 as follows
H r .Y; QY
H/
D 0; 8r < 0;
Hcr .Y; R€Y
H Q/
D 0; 8r > 0
and similarly for the second one. Moreover, by Theorem 5.1.1, if Y is affine, then
the vanishing results hold for every r ¤ 0.
Remark 5.1.4. It is essential that the embedding Y PN be affine. For example, the conclusion does not hold in the case when Y D A2 n f0g P2 and
Q D ZY Œ2:
5.2. Transversality, base change and choosing good flags. In this section we
highlight the role of transversality in the proof of Theorem 5.1.2. In fact, transversality implies several base change equalities which we use throughout the paper in
order to prove the vanishing results in Theorem 5.1.2, its iteration Lemma 6.1.1,
its “two-flag-extension” (27) and to observe (29). While the vanishing results are
used to realize condition (32), which is the key to the main results of this paper,
the base change equality (29) is used to reduce the results for the perverse spectral
sequences Theorems 4.1.3, 4.2.1 with respect to a map X ! Y , to analogous results
for perverse spectral sequences on Y .
These base change properties hold generically by virtue of the generic base
change theorem [11], and this is enough for the purposes of this paper. On the
other hand, it is possible to pinpoint the conditions one needs to impose on flags;
see Definition 5.2.4 and Remark 5.2.6.
Let Y PN be an affine embedding of the quasi projective variety Y and
S be the resulting projective completion. There is a natural decomposition
Y Y
`
`
S/
SnY /
into locally closed subsets PN D .PN n Y
.Y
Y: Let K 2 DY :
Definition 5.2.1 (Stratifications adapted to the complex and to the embedding).
S nY;
We say that a stratification † of PN is adapted to the embedding Y PN if Y; Y
N
S, and P n Y
S are unions of strata, and † induces by restriction stratifications
hence Y
N
N
S
S
S n Y with respect to which all possible inclusions among
on P ; P n Y ; Y ; Y; Y
these varieties are stratified maps. We denote these induced stratifications by †Y ;
etc. and we say that † is adapted to K if K is †Y -constructible.
THE LEFSCHETZ HYPERPLANE THEOREM
2101
Remark 5.2.2. Since maps of varieties can be stratified and a finite collection
of stratifications admits a common refinement, stratifications which are adapted to
the complex and to the embedding exist.
Let † be a stratification of PN adapted to K and to the embedding Y PN .
S D ƒ\Y
S. Set U
S WD .Y
SnH
S/ and
Let ƒ P be a hyperplane, H WD ƒ \ Y and H
U WD .Y n H /: Consider the cartesian diagram
(24)
i
H
/Y o
J
S
H
i
j
U
J
/Y
So
j
J
S:
U
We address the following question: when is the natural map
(25)
JŠ j j K ! j JŠ j K
S \ .Y
S n Y /: By the octahedron
an isomorphism? In general the two differ on H
axiom, the map (25) is an isomorphism if and only if the natural base change map
J i K ! i J K is an isomorphism. This latter condition is met if ƒ is general ([4,
Lemma 3.3]). In fact it is sufficient that ƒ meets transversally the strata in †YS Y .
This is a condition on the stratification, not on K. It follows that the analogous
map jŠ J J Š K ! J jŠ J Š K is also an isomorphism under the same conditions.
Proof of Theorem 5.1.2 (see [4]). We prove the first statement for cohomology.
The point is that a general linear section produces the isomorphism (25) and this
identifies the cohomology groups in question with compactly supported cohomology groups on affine varieties where one uses Theorem 5.1.1. Note that since the
maps of type j and J are affine, all the complexes appearing below are perverse.
We have the following chain of equalities:
S; J jŠ j Q/ D H r .Y
S; jŠ J j Q/;
H r .Y; jŠ j Q/ D H r .Y
S; jŠ J j Š Q/ D Hcr .U
S; J j Š Q/
D Hcr .Y
S is affine and J j Š Q is perverse, the last group is zero for r < 0
and, since U
and the first statement for cohomology follows. The one for compactly supported
cohomology is proved in a similar way.
In order to prove the second statement, we consider the Cartesian diagram
(26)
U \U0
j0
j
U0
j0
/U
j
/ Y:
2102
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
Since the embedding Y PN is affine, the open sets U; U 0 and U \ U 0 are affine.
Note that this fails if the embedding is not affine. We have that jŠ ; j ; j Š D j are
all t -exact and preserve perverse sheaves. The same is true for j 0 .
The equality jŠ j j0 j 0 Š Q D j0 j 0 Š jŠ j Q is proved using base change considerations similar to the ones we have made for (25).
We prove the vanishing in cohomology. The case of cohomology with compact supports is proved in a similar way. The case r < 0 is covered by the first
statement. We need suitable “reciprocal” transversality conditions which are the
obvious generalization of the ones mentioned in discussion of (25). We leave the
formulation of these conditions to the reader. It will suffice to say that they are met
by a general pair .ƒ; ƒ0 /: The case r > 0 follows from Theorem 5.1.1 applied to
the affine U 0 : H.Y; j0 j 0 Š jŠ j Q/ D H.U 0 ; j 0 Š jŠ j Q/:
Remark 5.2.3. Let Q 2 PY and † be a stratification of PN adapted to Y PN
Y
and such that Q 2 P†
Y : An inspection of the proof of Theorem 5.1.2 reveals that
S n Y: It is
it is sufficient to choose ƒ so that it meets transversally the strata in Y
not relevant how H meets the strata in †Y : A similar remark holds in the case of
a pair of hyperplanes.
We now introduce a kind of transversality notion that is sufficient for the
purpose of this paper. Let † be as above.
Definition 5.2.4 (Flag in good position wrt †). A linear n-flag F D ƒ0 ƒ 1 ƒ n on PN is in good position with respect to † if it is subject to the
following inductively defined conditions:
1) ƒ 1 meets all the strata of †0 WD † transversally; let † 1 be a refinement
of † such that its restriction to ƒ 1 ' PN 1 is adapted to the embedding
Y 1 ƒ 1;
2) ƒ 2 meets all the strata of † 1 transversally; we iterate these conditions and
constructions and introduce † 2 ; ƒ 3 , . . . , † nC1 ; ƒ n and we require that,
for every i D 1; : : : ; n;
i) ƒ i meets all the strata of † i C1 transversally.
We define †0 WD †
n:
Remark 5.2.5. By the Bertini theorem, it is clear that a general linear n-flag F
is in good position with respect to a fixed †: Of course, “general" depends on †:
Note also that if F is in good position, then Yp has pure codimension p in Y .
Remark 5.2.6. There is the companion notion of a pair of linear n-flags .F; F0 /
being in good position with respect to † and to each other. We leave the task of
writing down the precise formulation to the reader. The notion is again inductive and proceeds, also by imposing mutual transversality, in the following order:
ƒ 1 ; ƒ0 1 ; ƒ 2 ; ƒ0 2 , etc. It suffices to say that a general pair of flags will do.
2103
THE LEFSCHETZ HYPERPLANE THEOREM
The proof of Theorem 5.1.2 works well inductively with the elements of a
linear flag F on PN in good position with respect to the embedding and to the perverse sheaf Q. We use this fact in the proof of Lemma 6.1.1. Similarly, this kind of
argument works well with a pair of flags in good position with respect to the Q, the
embedding and each other. In particular, it works for a general pair of flags. In these
cases, we have the equality €Z j Z j 1 .QYi Yi 1 / D .€Z j Z j 1 Q/Yi Yi 1
which follows from repeated use of the equality jŠ j j0 j 0 Š Q D j0 j 0 Š jŠ j Q of
Theorem 5.1.2. By transversality, the shift Œi C j  of these complexes are perverse.
This allows us to apply the vanishing results of Theorem 5.1.2 and deduce, for
general pairs of flags on the quasi projective variety Y , that
(27)
H r .Y; €Z j Z j 1 .QYi Yi 1 //DHcr .Y; €Z j Z j 1 .QYi Yi 1 //D0; 8 r ¤ 0:
Let f W X ! Y be a map of varieties. The diagram (24) induces the cartesian
diagram:
(28)
XH
i
f
i
H
/X o
j
f
/Y o
j
XU
f
U:
The previous base change discussion implies, for ƒ meeting all strata of † transversally, that
(29)
f jŠ j C D jŠ j f C;
fŠ j j Š C D j j Š fŠ C:
Similar base change equations hold for a linear flag F on PN in good position with
respect to fŠ C and to the embedding Y PN (e.g. general) and also for a pair of
flags in good position with respect to fŠ C , the embedding and each other (e.g. a
general pair).
5.3. Two short exact sequences.
L EMMA 5.3.1. Let Y PN be quasi projective and Q 2 PY : If ƒ PN is a
general linear section, then there are natural exact sequences in
(30)
0 ! i i QŒ 1 ! jŠ j Š Q ! Q ! 0;
(31)
0 ! Q ! j j Q ! iŠ i Š QŒ1 ! 0:
Proof. There are the distinguished triangles in DY :
C
jŠ j Š Q ! Q ! i i Q !;
C
iŠ i Š Q ! Q ! j j Q ! :
Since j is affine, jŠ and j are t-exact and jŠ j Š Q and j j Q are perverse. We
choose ƒ so that it is transverse to the strata of a stratification for Q. It follows that
i i QŒ 1 D iŠ i Š QŒ1 is perverse. Each conclusion follows from the long exact
sequence of perverse cohomology of the corresponding distinguished triangles. 2104
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
5.4. The forget-the-filtration map. Let A be an abelian category. [5, Prop.
3.1.4.(i)] is a sufficient condition for the natural forget-the-filtration map
HomDF .A/ ! HomD.A/
to be an isomorphism. We need the bifiltered counterpart of this sufficient condition.
The objects .L; F; G/ of the bifiltered derived category DF 2 .A/ are complexes L endowed with two filtrations. The homotopies must respect both filtrations and one inverts bifiltered quasi isomorphisms, i.e. (homotopy classes of)
maps inducing quasi isomorphisms on the bigraded objects GraF GrbG : It is a routine
matter to adapt Illusie’s treatment of DF .A/ to the bifiltered setting and then to
adapt the proof of [5, Prop. 3.1.4.(i)] to yield a proof of
P ROPOSITION 5.4.1. Assume that A has enough injectives and that .L; F; G/,
.M; F; G/ 2 D C F2 .A/ are such that
j
n
HomDFA
..GriG LŒ i ; F /; .GrG M Œ j ; F // D 0;
8n < 0;
8i < j:
The “forget-the-second filtration” map is an isomorphism:
'
HomDF 2 A ..L; F; G/; .M; F; G// ! HomDFA ..L; F /; .M; F //:
5.5. The canonical lift of a t-structure. The following is a mere special case
of [4, App.]. Let A be an abelian category. The derived category D.A/ admits the
standard t-structure, i.e. usual truncation. The filtered derived category DF .A/
admits a canonical t-structure which lifts (in a suitable sense which we do not need
here) the given one on D.A/. This canonical t-structure on DF .A/ is described
as follows. There are the two full subcategories
DF .A/0 WD f.L; F / j GriF L 2 D.A/i g;
DF .A/0 WD f.L; F / j GriF L 2 D.A/i g:
The heart DF .A/0 \ DF .A/0 is
DF ˇ .A/ D f.L; F / j GriF LŒi  2 Ag;
where ˇ is for bête (see [5, 3.1.7]). The reader can verify the second axiom of t structure, i.e. Hom 1 .DFA0 ; DFA1 / D 0, by a simple induction on the length
of the filtrations, and the third axiom, i.e. the existence of the truncation triangles,
by simple induction on the length of the filtration coupled with the use of Verdier’s
“Lemma of nine” (see [5, Prop. 1.1.11]).
5.6. The key lemma on bifiltered complexes. Let A be an abelian category
and .L; P; F / be a bifiltered complex, i.e. an object in the bifiltered derived category DF 2 .A/. Recall the existence of the shifted filtration Dec.F / associated
with .L; F /.
THE LEFSCHETZ HYPERPLANE THEOREM
2105
The purpose of this section is to prove the following result, the formulation
of which has been suggested to us by an anonymous referee. This result is key to
the approach presented in this paper.
P ROPOSITION 5.6.1. Let .L; P; F / be a bifiltered complex such that
(32)
b
H r .GraF GrP
L/ D 0; 8r ¤ a
b:
Assume that L is bounded below and that A has enough injectives. There is a
natural isomorphism in the filtered derived category DF .A/,
.L; P / ' .L; Dec.F //
that induces the identity on L and thus identifies
.H .L/; P / D .H .L/; Dec.F //:
In particular, there is a natural isomorphism between the spectral sequences associated with .L; P / and .L; Dec.F // inducing the identity on the abutments.
In order to prove Proposition 5.6.1, we need the following two lemmata.
L EMMA 5.6.2. Let .L; F / be any filtered complex. Then the bifiltered complex .K; F; Dec.F // satisfies (32).
Proof. This is a formal routine verification.
L EMMA 5.6.3. With things as in Proposition 5.6.1, the natural map
HomDF 2 .A/ ..L; P; F /; .L; Dec.F /; F // ! HomDF .A/ ..L; F /; .L; F //
induced by forgetting the first filtration is an isomorphism. The same is true with
the roles of the filtrations P and Dec.F / switched.
Proof. Endow D.A/ with the standard t-structure (i.e. usual truncation). Endow DF .A/ with the canonical lift of this t-structure (see 5.5). The hypothesis
(32) implies that, for every b 2 Z;
b
.GrP
LŒ b; F / 2 DF ˇ .A/;
i.e., it is in the heart of the canonical lift of the standard t -structure to DF .A/:
Similarly, Lemma 5.6.2 implies that, for every b 2 Z;
.GrbDec.F / LŒ b; F / 2 DF ˇ .A/:
The hypotheses of Proposition 5.4.1 are met: in fact they are met for every
i; j , due to the second axiom of t-structure. The first statement follows.
If we switch P and Dec.F /; then the hypotheses of Proposition 5.4.1 are still
met, for the same reason, and the second statement follows.
2106
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
Proof of Proposition 5.6.1. By Lemma 5:6:3, the identity on .L; F / admits
natural lifts
P 2 HomDF 2 A ..L; P; F /; .L; Dec.F /; F //;
Dec.F / 2 HomDF 2 A ..L; Dec.F /; F /; .L; P; F //
which are inverse to each other and hence isomorphisms.
By forgetting the second filtration, we obtain a pair of maps in
HomDFA ..L; P /; .L; Dec.F // and HomDFA ..L; Dec.F //; .L; P /
which are inverse to each other. By forgetting both filtrations, both maps yield the
identity on L.
Remark 5.6.4. The results of this section hold if we replace Dec.F / with any
filtration P 0 satisfying (32).
6. Proof of the results
In this section, we prove the main results of this paper and make a connection
with Beilinson’s equivalence theorem [4].
6.1. Verifying the vanishing (32) for general flags. Recall the set-up: Y is
quasi projective of dimension n, K 2 DY , Y PN is an affine embedding, F; F0
is a pair of linear n-flags on PN : We have the bounded multi-filtered complexes of
abelian groups .R€.Y; K/; P; F; ı/; .R€c .Y; K/; P; G; ı/ obtained using suitably
acyclic resolutions. The perverse spectral sequences are the spectral sequences for
the filtration P , the flag spectral sequences are the ones for the filtrations F; G; ı,
similarly, for the perverse Leray spectral sequences. If the flags are arbitrary, then
the perverse and the flag spectral sequences seem unrelated.
Let † be a stratification of PN adapted to K and to the embedding Y N
P : The proof of Theorem 4.1.1 consists of showing that if the flag F is in good
position with respect to † (see Definition 5.2.4), then the vanishing conditions
(32) hold for the bifiltered complexes .R€.Y; K/; P; F // by virtue of a repeated
application of the strong weak Lefschetz Theorem 5.1.2, so that Proposition 5.6.1
applies and there is a natural identification of filtered complexes .R€.Y; K/; P / D
.R€.Y; K/; Dec.F // and of the ensuing spectral sequences. The other results are
proved in a similar way.
The key to the proof is Lemma 6.1.1 (below) which is suggested by a construction due to Beilinson [4, Lemma 3.3 and Complement to 3], which yields a
technique to construct resolutions of perverse sheaves on varieties by using suitably
transverse flags. The entries of the resolutions satisfy strong vanishing conditions
and realize the wanted condition (32). There are three versions, left, right and
bi-sided resolutions.
2107
THE LEFSCHETZ HYPERPLANE THEOREM
The resolutions are complexes obtained by using the following general conY
struction. Let Q 2 P†
Y , where †Y is the trace of † on Y . The connecting maps
C2 Q !
associated with the short exact sequences 0 ! GrC1
F Q ! F Q=F
Gr Q ! 0 give rise to a sequence of maps in DY
d
d
GrF n Q ! ! Gr0F Q;
(33)
with d 2 D 0: We call this a complex in DY . The same is true for the G filtration:
Gr0G Q ! ! GrnG Q is a complex in DY . The bigraded objects GrF GrG give
rise to a double complex with associated single complex Grı n Q ! ! Grnı Q in
DY : The transversality assumptions on the flags ensure that these are complexes of
perverse sheaves resolving Q, that they are suitably acyclic and that their formation
is an exact functor. More precisely, we have the following:
L EMMA 6.1.1 (Acyclic resolutions of perverse sheaves). Let Y be quasi proY
jective, † be a stratification adapted to the affine embedding Y PN , Q 2 P†
Y
be a †Y -constructible perverse sheaf on Y .
Let F be a linear n-flag on PN in good position with respect to †. Then
†0
(i) There is the short exact sequence in PY Y PY W
0 ! QY
n
n ! : : : ! QY
∅Œ
1
Y
Œ 1 ! QY
2
(i0 ) If , in addition, Y is affine, then H r .Y; QYp
Yp
†0Y
(ii) There exists the short exact sequence in PY
0 ! Q ! k0 k0Š Q ! k
Š
1 k 1 QŒ1
1
1
Œ0 ! Q ! 0I
/ D 0; 8r ¤ p:
PY :
! ::: ! k
(ii0 ) If , in addition, Y is affine, then Hcr .Y; k
Y
Š
p k p Q/
Š
n k n QŒn
! 0I
D 0; 8r ¤ p:
Let .F; F0 / be a pair of linear n-flags which are in good position with respect to †
and to each other. Then
(iii) The single complex Grı QŒ associated with the double complex of perverse
j
sheaves GriF GrG QŒi C j  is canonically isomorphic to Q in D b .PY /:
(iii0 ) H r .Y; .k 0 j k 0 Š j Q/Yi
r ¤ i Cj.
Yi
1
/ D Hcr .Y; .k 0
j k
0Š
j Q/Yi Yi
1
/ D 0 for every
Proof. Note that, in Lemma 5.3.1,
Y D Y0 ;
jŠ j Q D k0Š k0 Q D QY0
Y
1
;
and j j Š Q D k0 k0Š Q:
More generally,
R€Z
j
Z
j
1
D k0
j k
0Š
j
and
. /Yi
Yi
1
D ki Š ki :
2108
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
Statement (i) follows by a simple iteration of Lemma 5.3.1, where one uses at
each step the fact that F is in good position with respect to the initial †: In this step,
the relative position of the linear sections and the strata at infinity are unimportant.
Statement (i0 ) follows from an iterated use of Theorem 5.1.2 and Remark 5.2.3.
Here it is important that the linear sections meet the strata at infinity transversally.
Statements (ii) and (ii0 ) are proved in a similar way.
The double complex is obtained as follows: first resolve Q as in (ii), then
resolve each resulting entry as in (i). We thus have quasi isomorphisms in C b .PY /:
Q ! GrG QŒ Grı QŒ and (iii) follows. Finally, (iii0 ) now follows from (27).
†
PY Y
Remark 6.1.2. The formation of the left, right and bi-sided resolutions of Q 2
in Lemma 6.1.1 are exact functor with values in C b .PY /:
Assumption 6.1.3 (Choice of the pair of linear flags F; F0 ). We fix a pair of
linear n-flags F; F0 on PN in good position with respect to † and to each other. A
general pair in F .N; n/ F .N; n/ will do.
Remark 6.1.4. Since K 2 DY†Y ; the perverse sheaves pHb .K/ 2 DY†Y and,
with our choice of the pair .F; F0 /, the conclusions of Lemma 6.1.1 hold for all the
pHb .K/.
L EMMA 6.1.5. If Y is affine, then
b
H r .GraF GrP
R€.Y; K// D 0;
8r ¤ a
b;
b
H r .GraG GrP
R€c .Y; K// D 0;
8r ¤ a
b:
If Y is quasi projective, then
b
b
H r .Y; Graı GrP
R€.Y; K// D H r .Y; Graı GrP
R€c .Y; K// D 0
8r ¤ a
b:
Proof. We prove the first assertion and the second and third are proved in a
similar way. By (12), the group in question is
Hr
.a b/
.Y; pH
b
.K/Ya
Ya
1
Œa/
and the required vanishing follows from Assumption 6.1.3, Remark 6.1.4, and
Lemma 6.1.1(i0 ).
6.2. Proofs of Theorems 4.1.1, 4.1.2, 4.1.3 and 4.2.1.
Proof of Theorems 4.1.1 and 4.1.2. By the first two assertions of Lemma 6.1.5,
we can apply Proposition 5.6.1 to .R€.Y; K/; P; F / and to .R€c .Y; K/; P; G/ and
prove the first two theorems.
Proof of Theorem 4.1.3. We prove the version for H.X; C /. The case of
Hc .X; C / is proved in a similar way. Given the fixed embedding Y PN , pick
THE LEFSCHETZ HYPERPLANE THEOREM
2109
a stratification † of PN adapted to f C and to the embedding. Choose a linear
n-flag F on PN in good position with respect to †, e.g. general. Let Y be the
corresponding n-flag on Y and set X WD f 1 Y : Denote by zi ; jz; kz the associated
embeddings as in (4).
By Theorem 4.1.1, the perverse spectral sequence for H.Y; f C /, i.e. the perverse Leray spectral sequence for H.X; C /, is the shifted Y flag spectral sequence
for H.Y; f C / (F -version).
Our goal is to identify the Y flag spectral sequence for H.Y; f C / with the
X spectral sequence for H.X; C /. It is sufficient to show that
(34)
.R€.X; C /; FX / D .R€.Y; f C /; FY /I
in fact, the shifted versions would also coincide and we would be done. In general,
the two filtered complexes for Y and X do not coincide, due to the failure of
the base change theorem. In the present case, transversality prevents this from
happening.
We assume that C is injective. The filtered complex .C; F / is of c-soft type.
On varieties c-soft and soft are equivalent notions and soft sheaves are fŠ and
f -injective.
We have the filtered complex .R€.X; C /; FX /, i.e. the result of applying
€.X; / to the C -analogue of (5).
Transversality ensures that we have the first equality in (29): jp Š jzp f C D
f jp Š jp C: This implies that, by applying f to the C -analogue of (5), we obtain
the f C analogue of (5) on Y with respect to Y , i.e. (34) holds and we are done.
Proof of Theorem 4.2.1. In view of the third assertion of Lemma 6.1.5, the
proof is analogous to the proofs given above.
6.3. Resolutions in D b .PY /. In an earlier version of this paper, we worked
in the derived category of perverse sheaves D b .PY / which, in the case of field
coefficients, is equivalent to DY ([4]). We are very thankful to one of the anonymous referees for suggesting the considerably more elementary approach contained
in this paper which takes place in D.Ab/. On the other hand, the approach in
D b .PY / explains the relation “P D Dec.F /” at the level of complexes of (perverse)
sheaves, i.e. before taking cohomology. We outline this approach in the case of the
F -construction on Y affine. We omit writing down the similar details in the case
of the G-construction in the affine case and in the case of the ı.F; G/-construction
in the quasi projective case.
The approach is based on Beilinson’s Equivalence Theorem [4].
In what follows, Y is affine; we work with field coefficients, for example Q,
b
D .PY / is endowed with the standard t-structure, DY with the perverse t -structure.
2110
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
An equivalence of t-categories is a functor, between triangulated categories with tstructures, which is additive, commutes with translations, preserves distinguished
triangles, is t -exact (i.e. it preserves the hearts) and is an equivalence.
T HEOREM 6.3.1 ([4]). There is an equivalence of t-categories, called the realization functor
'
rY W D b .PY / ! DY :
An outcome of this result is that it implies that, up to replacing K 2 DY
with a complex naturally isomorphic to it, there is a filtration B on K such that
b
GrB
KŒb 2 PY . When we recall Section 5.5, this means that .K; B/ is in the
heart DY Fˇ of the canonical lift to DY F of the perverse t -structure on DY . This
circumstance, coupled with the construction (33), allows us to describe an inverse
sY to rY , i.e. to assign to K a complex of perverse sheaves
sY .K/ D sY .K; B/ D GrB
KŒ DW K 2 D b .PY /:
b
Fix a stratification S of Y such that all the finitely many nonzero GrB
K are
0
0
b
†
†
S-constructible. Note that if K 2 DY , then it is possible that GrB K … D and
one may need to refine. Choose an embedding Y PN , a stratification † on PN
adapted (cf. Definition 5.2.1) to S and to the embedding, and a linear n-flag F on
PN in good position (cf. Definition 5.2.4; a general one will do) with respect to †.
Let  D .F; P / be the diagonal filtration. By transversality, we have that
.K; / 2 DY Fˇ : We obtain the double complex K; WD GrF GrP
K, with asso;
ciated single complex s.K / D sY .K; / that maps quasi isomorphically onto
K : We also have H r .Y; K; / D 0 for every r ¤ 0 so that we have obtained a
resolution with H.Y; /-acyclic entries.
The single complex s.K; / admits the bête filtration by rows Brow , where
q
Brow s.K; / is the single complex associated with the double complex K;q .
i.e. the result of replacing with zeroes the entries strictly above the q-th row.
p
There is another filtration, Stdcol ; where Stdcol s.K; / is the single complex
associated with the double complex obtained by keeping the columns p 0 < p;
replacing the columns p 0 > p with zeroes, and replacing the entries Kp;q in the
. p/-th column by KerfKp;q ! KpC1;q g:
By Remark 6.1.2, the exactness properties of the construction of the resolution of Lemma 6.1.1 ensure that the natural map .s.K; /; Stdcol /!.K ; Std/ is a
filtered quasi isomorphism.
It is via this construction that the relation Dec.F / D P becomes transparent:
it holds in D b .PY / ' DY and it descends to D.Ab/:
1) It is elementary to verify that Dec.Brow / D Stdcol (cf. [3, Rem. 3.11.1]);
2) The filtered complex of perverse sheaves .s.K; /;Brow / corresponds to .K;F /
under the equivalence rY ;
THE LEFSCHETZ HYPERPLANE THEOREM
2111
3) The t -exactness of rY ensures that the filtered complex .s.K; /; Stdcol / '
.K; Std/ corresponds to .K; P /;
4) By the exactness of the construction, the complex s.H 0 .Y; K; // inherits the
relation Dec.Brow / D Stdcol .
5) The bifiltered complex .s.H 0 .Y; K; //; Stdcol ; Brow / is a realization in the
bifiltered derived category DF2 .Ab/ of .R€.Y; K/; P; F / and by 4) the perverse spectral sequence is identified with the shifted flag spectral sequence.
Remark 6.3.2. On the affine Y , the functor HP0Y W PY ! Ab, Q 7! H 0 .Y; Q/
is right-exact. By [4, §3], this right-exact functor admits a left-derived functor
LHP0Y W D .PY / ! D .Ab/. The complex in Step 4) realizes LHP0Y .K/.
7. Applications
The following results are due to M. Saito [18] who used his own mixed Hodge
modules. We offer a proof based on the methods of this paper.
T HEOREM 7.0.3. Let Y be quasi projective. The perverse spectral sequences
p;q
D H 2pCq .Y; pH
p
p;q
D Hc2pCq .Y; pH
p
E1
E1
.ZY // H) H .Y; Z/;
.ZY // H) Hc .Y; Z/
are spectral sequences in the category of mixed Hodge structures.
Proof. We prove the first statement when Y is affine. The other variants are
proved in similar ways. By Theorem 4.1.1, there is an n-flag Y on Y such that
the perverse spectral sequence for H.Y; Z/ is the shifted spectral sequence of the
flag spectral sequence
p;q
E1
D H pCq .Yp ; Yp
1 ; Z/
H) H .Y; Z/;
which is in the category of mixed Hodge structures.
T HEOREM 7.0.4. Let f W X ! Y be a map of varieties where Y is quasi
projective. The perverse Leray spectral sequences
p;q
E1
p;q
E1
D H 2pCq .Y; pH
D Hc2pCq .Y; pH
p
.f ZY // H) H .X; Z/;
p
.fŠ ZY // H) Hc .X; Z/
are spectral sequences in the category of mixed Hodge structures.
Proof. We prove the case of cohomology over an affine base Y and leave
the rest to the reader. By Theorem 4.1.3, the perverse Leray spectral sequence for
H.X; Z/ is the shifted X flag spectral sequence with respect to a suitable n-flag
X on X . This latter is in the category of mixed Hodge structures.
2112
MARK ANDREA A. DE CATALDO and LUCA MIGLIORINI
Remark 7.0.5 (Mixed Hodge structures and the decomposition theorem). In
the paper [8], we endow the cohomology of the direct summands, appearing in the
decomposition theorem for the proper push forward of the intersection cohomology
complex of a proper variety, with natural, pure polarized Hodge structures. These
structures arise as subquotients of the pure Hodge structure of the cohomology
of a resolution of the singularities of the domain of the map. In particular, this
endows the intersection cohomology groups of proper varieties with pure polarized
Hodge structures. In the paper [9], we prove that for projective morphisms of
projective varieties, one can realize the direct sum splitting mentioned above in the
category of pure Hodge structures. The methods of this paper allow us to endow
the intersection cohomology groups IH.Y; Z/ and IHc .Y; Z/ of a quasi projective
variety with a mixed Hodge structure and to extend all the results of [8] to the
case of quasi projective varieties. Furthermore, we compare the resulting mixed
Hodge structures with the ones arising from M. Saito’s work and we show that they
coincide. Details will appear in [6].
Acknowledgments. It is a pleasure to thank D. Arapura, A. Beilinson, M.
Goresky, M. Levine, M. Nori for stimulating conversations. The first author thanks
the University of Bologna and I.A.S. Princeton for their hospitality during the
preparation of this paper. The second author thanks the Centro di Ricerca Matematica E. De Giorgi in Pisa, the I.C.T.P., Trieste, and I.A.S., Princeton for their
hospitality during the preparation of this paper. Finally, we thank the referees for
pointing out several inaccuracies in an earlier version of the paper, and for very
useful suggestions on how to make the paper more readable.
The first-named author dedicates this paper to Caterina, Amelie and Mikki.
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(Received May 21, 2007)
(Revised May 29, 2008)
E-mail address: [email protected]
M ATHEMATICS D EPARTMENT , S TONY B ROOK U NIVERSITY , S TONY B ROOK , NY 11794-3651,
U NITED S TATES
E-mail address: [email protected]
D EPARTMENT OF M ATHEMATICS , U NIVERSITÀ DI B OLOGNA , P IAZZA DI P ORTA S. D ONATO , 5,
B OLOGNA , I TALY